Problem 24 Solve each equation by using the... [FREE SOLUTION] (2024)

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Chapter 8: Problem 24

Solve each equation by using the quadratic formula. $$-x^{2}-3 x+5=0$$

Short Answer

Expert verified

The solutions are \[ x = -\frac{3 + \sqrt{29}}{2} \] and \[ x = -\frac{3 - \sqrt{29}}{2} \]

Step by step solution

Write down the quadratic formula

The quadratic formula is given by \[ x = \frac{-b \,\pm\, \sqrt{b^{2} - 4ac}}{2a} \]where the equation is of the form \[ ax^{2} + bx + c = 0 \]

Identify coefficients a, b, and c

For the equation \[ -x^{2} - 3x + 5 = 0 \]a = -1, b = -3, and c = 5.

Plug the coefficients into the quadratic formula

Substitute the coefficients into the quadratic formula: \[ x = \frac{-(-3) \,\pm\, \sqrt{(-3)^{2} - 4(-1)(5)}}{2(-1)} \]This simplifies to \[ x = \frac{3 \,\pm\, \sqrt{9 + 20}}{-2} \]

Simplify under the square root

Combine the terms inside the square root: \[ x = \frac{3 \,\pm\, \sqrt{29}}{-2} \]

Split into two solutions

The solutions are: \[ x = \frac{3 + \sqrt{29}}{-2} \] and \[ x = \frac{3 - \sqrt{29}}{-2} \]

Simplify each solution

Simplify each fraction: \[ x = -\frac{3 + \sqrt{29}}{2} \] and \[ x = -\frac{3 - \sqrt{29}}{2} \]

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving quadratic equations

Solving quadratic equations can be done using the quadratic formula. A quadratic equation is any equation that can be written in the form . When you have this standard form, you can use a special formula to find 'x', called the quadratic formula. The formula is . . Plug the coefficients from your equation into this formula to solve for 'x'. You will end up with two potential solutions, because of the plus and minus sign ( ). Breaking down the process into steps can make it much easier to manage!

identifying coefficients

Coefficients are the numbers in front of the variables in your quadratic equation. For example, in the equation , we have: . The number in front of is your ('' here is -1), the number in front of is your ('' here is -3), is your ('' here is 5). Once you identify , and , you can substitute these values into the quadratic formula. Identifying the coefficients correctly is crucial for solving the equation accurately.

simplifying expressions

Simplifying expressions is a key step when using the quadratic formula. For instance, just substituting the coefficients into the formula isn't enough; you must also simplify under the square root and the resulting fractions. Here is an example using the equation : . Next, simplify under the square root: + . After simplifying: . Finally, split and simplify each solution: and . Simplifying makes it easier to understand and use your solutions.

roots of quadratic equations

The roots of a quadratic equation are the values of that satisfy the equation. When you solve a quadratic equation using the quadratic formula, you will end up with two values for because of the part of the formula. These values are called the roots. For the equation , after identifying the coefficients and simplifying, you get the roots: and . These roots tell you where the parabola (the graph of the quadratic equation) intersects the -axis.

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Most popular questions from this chapter

Solve each inequality. State the solution set using interval notation whenpossible. $x^{2} \geq 36$Graph each of the following equations by solving for $y$ a) $x=y^{2}-1$ b) $x=-y^{2}$ c) $x^{2}+y^{2}=4$The equation $x=y^{2}$ is equivalent to $y=\pm \sqrt{x} .$ Graph both$y=\sqrt{x}$ and $y=-\sqrt{x}$ on a graphing calculator. How does the graph of$x=y^{2}$ compare to the graph of $y=x^{2} ?$Find the exact solution $(s)$ to each problem. If the solution(s) areirrational, then also find approximate solution(s) to the nearest tenth. More missing numbers. Find two real numbers that have a sum of 8 and a productof 2Find $b^{2}-4 a c$ and the number of real solutions to each equation. $$9-24 z+16 z^{2}=0$$

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